Fixed-income attribution

Overview

Fixed-income attribution is the process of measuring returns generated by various sources of risk in a fixed income portfolio, particularly when multiple sources of return are active at the same time.

Importance

The risks affecting the return on a bond portfolio, as an example, include the overall level of the yield curve, the slope of the yield curve, and the credit spreads of the bonds in the portfolio. A portfolio manager may hold firm views on the ways in which these factors will change in the near future, so in three separate risk decisions he positions the assets in the portfolio to take advantage of these expected market movements. If all views subsequently prove to be correct, then each decision will generate a profit. If one view is wrong, it will generate a loss, but the effect of the other bets may compensate. The overall performance will then be the sum of the performance contributions from each source of risk. Attribution is therefore an extremely useful tool in verifying a fund manager's claims to possessing particular investment skills. If a fund is marketed as being interest-rate neutral while providing consistent returns from superior credit research, then an attribution report will confirm this claim. Conversely, if the attribution report shows that this same manager is making non-zero returns from interest rate movements, then his exposure to interest rate risk is clearly not zero and his investment process clearly differs from his stated position. Fixed-income attribution therefore provides a much deeper level of information than is available from a simple portfolio performance report. Typically, such a report only shows returns at an aggregated level, and provides no feedback as to where the investor's true skills lie. For these reasons, fixed-income attribution is rapidly growing in importance in the investment industry; see Financial risk management § Investment management.

Sector-based attribution

Among the simplest fixed income attribution techniques is sector-based attribution. This is based on the standard Brinson-Fachler attribution scheme, where the securities in the portfolio and benchmark are divided up into buckets based on their modified duration. This scheme has the advantage that it is readily understandable, particularly by managers who have an equity background. However, it does not provide a very deep analysis. The overall effects of a parallel change in the yield curve are supplied but there is none of the more detailed analysis supplied by a true fixed-income decomposition. A useful account of sector-based attribution, with worked examples, is provided in Dynkin et al. (1998).

Yield curve attribution

A more widely used approach to fixed-income attribution is to decompose the returns of individual securities by source of risk, and then to aggregate these risk-specific returns over an entire portfolio. Typical sources of risk include yield return, return due to yield curve movements, and credit spread shifts. These sub-returns can then be aggregated over time and sector to give the overall portfolio return, attributed by source of risk.

Sources of return

Over a given interval, the return of each security will be made up of return from various sub-returns (see below for explanations)

return due to yield (equivalently coupon, or accrued interest, or running yield); return due to rolling down the yield curve; return due to movements in the reference yield curve; return due to credit shifts; other sources of return, such as option-adjusted spread (OAS), liquidity, inflation, paydown, etc.

First principles versus perturbational attribution

To calculate the return arising from each effect, we can reprice the security from first principles by using a pricing formula, or some other algorithm, before and after each source of return is considered. For instance, in calculating yield return, we might calculate the price of the security at the start and end of the calculation interval, but using the yield at the beginning of the interval. Then the difference between the two prices may be used to calculate the security's return due to the passage of time. This approach is simple in principle but can lead to operational difficulties. It requires

accurate pricing formulae including, where relevant, ex-coupon, settlement, and country-specific conventions; security-specific data, such as day-count conventions and whether a bond has a non-standard first and last coupons; accurate inputs to these formulae, including market yields and other variable quantities such as the 90-day bank bill swap rate (BBSW) and consumer price index (CPI) factors for floating rate notes and inflation-linked securities, and regular updates for these quantities; a reconciliation function between existing performance measurement systems and the attribution system For these reasons, a pricing model-based approach to attribution may not be the right one where data sourcing or reconciliation is an issue. An alternative solution is to perform a Taylor expansion on the price of a security

P

(

y , t

)

{\displaystyle P\left({y,t}\right)}

and remove higher-order terms, which gives

δ P =

∂ P

∂ t

δ t +

∂ P

∂ y

δ y +

1 2

∂

2

P

∂

Modeling the yield curve

Historically, one of the most important drivers of return in fixed-income portfolios has been the yield curve, and many investment strategies are expressed in terms of changes in the curve. Any discussion of fixed-income attribution therefore requires an appreciation of how changes in the curve are described, and their effect on the performance of a portfolio. If one is only interested in gross changes in the yield curve at a particular maturity, then one can read yields off the various datasets, using interpolation where necessary, and there is no need to model any part of the curve. If, on the other hand, one wants to describe curve movements in terms used by traders (or to extrapolate), then some form of parameterization is required. The most widely used nomenclature for describing yield curve changes uses the terms "shift", "twist" and "butterfly". Briefly:

shift measures the degree to which a curve has moved upwards or downwards, in parallel, across all maturities twist measures the degree to which the curve has steepened or flattened. For instance, one might measure the steepness of the Australian yield curve as the difference between the 10-year bond future yield and the 3-year bond future yield. curvature (or butterfly, or curve reshaping) measures the degree to which the term structure has become more or less curved. For instance, a yield curve that can be fitted to a straight line exhibits no curvature at all. Describing these movements in numerical terms typically requires fitting a model to the observed yield curve with a limited number of parameters. These parameters can then be translated into shift, twist, and butterfly movements – or whatever other interpretation the trader chooses to use. This model is often also used for extrapolating credit default swaps (CDS). Two of the most widely used models are polynomial functions and Nelson-Siegel functions (Nelson and Siegel (1987)).

Here, polynomial functions are usually of the form

y

( m )

=

a

0

+

a

1

m +

a

2

m

2

{\displaystyle y\left(m\right)=a_{0}+a_{1}m+a_{2}m^{2}}

where

m

{\displaystyle m}

Factor-based attribution

A factor-based model of yield curve movements is calculated by deriving the covariance matrix of yield shifts at predefined maturities, and calculating the eigenvectors and eigenvalues of this matrix. Each eigenvector corresponds to a fundamental model of the yield curve, and each eigenvector is orthogonal, so that the curve movement on any given day is a linear combination of the basis eigenvectors. The eigenvalues of this matrix then give the relative weights, or importance, of these curve shifts. [Phoa (1998)]. Factor models use a large sample of historical yield curve data and construct a set of basis functions that can be linearly combined to represent these curve movements in the most economical way. The algorithm always attributes as much of the curve movement to the first basis function, then as much as possible to the second, and so on. Since these functions roughly correspond to our shift and twist motions, this approach attributes almost all of the curve change to these two modes, leaving a very small contribution from higher modes. Typical results attribute 90% of curve movements to shift changes, 8% to twist, and 2% to curvature (or butterfly) movements. However, the issue that these basis functions may be different from those in which the risk decisions were expressed is not widely appreciated. Since conventional risk analysis for fixed-income instruments usually assumes a parallel yield shift across all maturities, it would be most convenient if a parallel motion mode turned out to dominate the other modes, and in fact this is more or less what occurs. While a factor-based decomposition of term structure changes is mathematically elegant, it does have some significant drawbacks for attribution purposes:

Firstly, there is no agreement as to what these fundamental modes actually are, since they depend on the historical dataset used in the calculation (unlike, say, a parallel curve shift – which may be defined in purely mathematical terms). Each market, over each analysis interval, will therefore produce a different set of fundamental modes and hence different attribution decompositions, and so it may be impossible to compare sets of attribution results over longer intervals. By deciding to use such an approach, one is implicitly locked into a particular data history and (in practice) data/software vendor. The shape of the modes may not match user expectations, and in practice it will be most unlikely that the portfolio will be managed and hedged with reference to these fundamental modes. A manager is more likely to view future curve movements in terms of a simple shift and twist. The great advantage of a factor-based approach is that it ensures that as much curve movement as possible is attributed to shift movement, and that twist and curvature motion are given as small values as possible. This allows apparently straightforward reporting, because hard-to-understand curve movements are always assigned small weights in an attribution analysis. However, this is at the cost of a distortion of the other results. On the other hand, a naïve interpretation of the terms shift, twist, curvature when applied to yield curve movements may well give rise to higher-order movements that are much higher than investors would expect. There are also problems in the exact definition of the terms shift and twist. Without fixing a twist point at the outset, there is no unique value for these terms in either a Nelson-Siegel or polynomial formulation. However, the location of this twist point may not match user expectations.

Interest returns

The first source of return in a fixed-income portfolio is that due to interest. The majority of securities will pay a regular coupon, and this is paid irrespective of what happens in the marketplace (ignoring defaults and similar catastrophes). For instance, a bond paying a 10% annual coupon will always pay 10% of its face value to the owner each year, even if there is no change in market conditions. However, the effective yield on the bond may well be different, since the market price of the bond is usually different from the face value. Yield return is calculated from

r

y i e l d

= y ⋅ δ t

{\displaystyle r_{yield}=y\cdot \delta t}

where

y

{\displaystyle y}

is the security's yield to maturity, and

δ t

{\displaystyle \delta t}

is the elapsed time. Towards the end of the bond's life we often see a pull-to-parity effect. As maturity approaches, a bond's price converges to its nominal amount, irrespective of the level of interest rates, and this may cause a bond's price to move in a different way to what would normally be expected.

Roll return

Roll return can occur when a yield curve is steeply sloped. In the absence of any changes in the curve, as a security is held over time its maturity will decrease and the yield (as read off the curve) will change. If the slope is positive, the yield will decrease and the security's price will increase. Positioning a portfolio's assets to take advantage of a steeply sloping yield curve is sometimes called riding the yield curve. Strictly speaking, roll return belongs in a separate category, as it is neither a strict yield effect nor a return caused by a change in the yield curve.

Yield curve attribution

Changes in term structure form one of the most important sources of risk in a portfolio. Unlike an equity price, which just moves one-dimensionally, the price of a fixed-income security is calculated from sum of discounted cash flows, where the discount rate used depends on the interest rate at that maturity. The magnitude and shape of curve changes are therefore of major importance to fixed-income managers. At the most basic level, we can break down yield changes in terms of treasury shift and credit shift. At any maturity, we can compare the change in the target security with the change in the corresponding government-backed security, which will have the highest credit rating and hence the lowest yield. All securities have yields equal or greater than their equivalent-maturity government securities, which act as a benchmark for movements in the marketplace. Many investment-grade securities are traded at a spread to the Treasury curve, with the size of this spread depending on current economic conditions and the credit rating of the individual security. For instance, in April 2005 General Motors debt was downgraded to non-investment, or junk, status by the ratings agencies. As a result, the credit spread (or return demanded by investors for holding this riskier investment) rose by over 150 basis points, and the value of General Motors bonds accordingly fell. The loss in performance this caused was attributed entirely due to credit effects. Since the yield of virtually any fixed-income instrument is affected by changes in the shape of the Treasury curve, it is not surprising that traders examine future and past performance in the light of changes to this curve.

Appropriate yield curves

It is not always appropriate to use a single yield curve throughout a portfolio, even for instruments traded from a particular country. Inflation-linked securities use their own curve, whose movements may not show strong correlation with the yield curve of the broader market. Short-term money market securities may be better modeled by a separate model for the bill curve, and other markets may use the swap curve rather than the treasury curve.